As a result of the constant progress of the society and the development of science and technology, complex phenomena in complex networks have received more attention [1–9] in the last decades. The Small-World model [4] (Watts and Strogatz, 1998) and the BA scale-free network model [5] (Barabási and Albert 1999) represented well-known landmarks and contributions in the last century. After that, the research about complex network is rising. One of the most important trends of research in the field is that of cascading failures. A cascade failure is a particular type of system failure, one or more damaged nodes (or edges) lead other ones to consequent failures as a result of coupling relationship between nodes (or edges), resulting in the collapse of the whole network. Considering the great harm of cascading failures, many researchers established models based on real networks to put forward preventive and controlled method.

The vast majority of researches on cascading failures have concentrated exclusively on isolated networks, ignoring the fact that many real-world networks interact with each other. An example of this tendency is represented by the framework of analysis of cascading failures in interdependent networks [10–15] proposed by Buldyrev et. al. [15].

They found threshold value p_c is subjected to degree distribution. In single network, broader degree distribution causes smaller p_c. But in interdependent networks, broader degree distribution causes bigger p_c, because high-degree nodes of one network can depend on low-degree nodes of the other. In a different perspective, Gao et al. [16] studied the dynamics of the cascades of failures in n coupled interdependent networks under a random initial attack on one of the networks. And then they found, for any tree-like network of networks, the critical percolation threshold and the mutual giant component depend only on the number of networks but not on the topology. In the same token, Parshani et al. [17] showed that reducing the coupling between interdependent networks leads to a change from the first order percolation phase transition to the second order percolation transition at a critical point. In consonance with these lines of research, Huang et al. [18] studied the robustness [19–24] of interdependent networks under tunable degree-targeted attacks. They found that interdependent networks are difficult to defend by strategies such as protecting the high degree nodes. In the same vein, Schneider et al. [25] proposed a systematic strategy for selecting a minimum number of autonomous nodes that reduce the chances of a catastrophic cascading failure. Wang et al. [26] introduced load, load redistribution, and node capacity to interdependent networks, and investigated the robustness of interdependent networks with different inter-linkages. They found that the different inter-linkages have a dramatic effect on the robustness of interdependent networks.

Their researches are based on coupled networks, in which two subnets are interdependent. But another kind of coupled networks is rarely studied, in which two subnets have the same function, and we call it functionally identical coupled networks. An example of functionally identical coupled networks is two interconnected grids. In America, the grid comprises many independent grids and many power transmission lines intended to interconnect them. Therefore, study on robustness of functionally identical coupled networks is important. Therefore, in this work, two power grids are considered as two independent networks and power transmission lines as the interlinks connecting them. We propose this configuration as functionally identical coupled networks and we aim at studying its robustness.

The interdependent networks and the functionally identical coupled networks are fundamentally different, by two main reasons.

(1) The functions of two sub-networks in interdependent networks are interdependent. But in functionally identical coupled networks, they are the same. (2) In interdependent networks, a failed node in one of them causes a failure in the corresponding node of the other network. But in functionally identical coupled networks, the only reason a node crashes is that it cannot handle the extra load from nodes directly connected to it.

The rest of this paper includes these sections: Cascading failures model is introduced in section 2. And in section 3, we show the experimental results and analysis of robustness. Finally, in section 4, we summarize our paper.

The aims of this paper are to study the effect of the broadness and average degree of the degree distribution on robustness in functionally identical coupled networks and single network under random attack. We compare the robustness between SF-SF, SF-ER and ER-ER, and then help to construct robust functionally identical coupled networks.

For the sake of simplicity, for two networks A and B with the same size N, A_i (1≤i≤N) denotes a node in network A, and B_i (1≤i≤N) denotes a node in network B. The degree kAi of node A_i is the number of nodes connected to A_i in network A; and the degree kBi of node B_i is the number of nodes connected to B_i in network B. Every node in network A links only one node in network B and vice-versa. We assume that node A_i is linked to node B_i in network B and vice-versa.

In every network, each node has a load and a finite capacity, in the same way that in a power grid each power station has a load and a finite capacity. The node’s load corresponds to the intensity of electric current flowing in a power station, whereas its capacity corresponds to the biggest load that a power station can handle before breaking down. Generally, a higher-degree node carries a higher load. We assume that the initial load of a node is equal to its degree, and that its capacity is proportional to its initial load [23]. Consequently, initial load of node A_i in network A is defined as LAi=kAi and initial load of node B_i in network B as LBi=kBi. Similarly, node’s A_i capacity in network A is defined as CAi=(1+β)kAi and node’s B_i capacity in network B as CBi=(1+β)kBi, where β is the tolerance parameter, and β>0 [23].

The strategy that we propose here consists in randomly removing one node from functionally identical coupled network. For example, consider that a node A_i is removed. Then, its load will be redistributed among its neighboring nodes. However, the new extra-load on these neighboring nodes may exceed their capacity, causing them to fail. These new failed nodes provoke the new load redistribution. After several iterations, network is divided into many clusters. In the same way that in a power grid, if one power station is failed, electricity demand of the region will be supported from the surrounding power station connected to the failed one. Load redistribution may lead cascading failures.

We define the set of neighboring nodes of A_i as ΓAi, which consists of node B_i in network B, and neighboring nodes of A_i in network A. We assume that the extra load received by a node in ΓAi from A_i is proportional to its degree. In consequence, the extra load ΔLAjAi received by the neighboring node A_j(Aj∈ΓAi) in network A is: ΔLAjAi=LAikAj/∑Aa∈ΓAikAa, and the extra load ΔLBiAi received by neighboring node B_i(Bi∈ΓAi) in network B ΔLBiAi=LAikBi/∑Aa∈ΓAikAa.

The process of cascading failures is illustrated in Fig 1. Assuming that the cascading failures are induced by removing node i, and that the size of normal nodes in functionally identical coupled networks is S_i after a cascade failure, we use S = ∑_i∈A∪Bs_i/(2N)² to characterize the robustness of functionally identical coupled networks.

We construct three types of functionally identical coupled networks: the ER-ER network (Erdös-Rényi random network [6–7]), the SF-ER network and the SF-SF (SF is BA scale-free network [5], which is generated by an algorithm proposed by Dorogovtsev [27]).

Our model is implemented by Java programming language running on a PC with a 3.1 GHz processor, 6.0 GB memory and Windows 7 operating system. Data of the western United States power grid (S1 File) used in Part 3.3 was compiled by Duncan Watts and Steven Strogatz [4].

Many researchers study the robustness based on interdependent networks, in which each node depends on another. But few people consider the function consistency of coupled networks. In this paper, we fill this gap and study the robustness of functionally identical coupled networks against cascading failures under random attack. In section 3.1, we find that the SF-SF or ER-ER is more robust than SF or ER under random attack. In simple terms, in order to improve the robustness of two single SF or ER networks under random attack, we should couple them to SF-SF or ER-ER networks. We also can divide a single SF or ER network into a SF-SF or ER-ER networks for better robustness. Then, increase of λ and the average degree enhances the robustness in SF-SF and SF networks. In section 3.2, Fig 4 tells us that we should couple two single networks as SF-SF for stronger robustness instead of SF-ER or ER-ER. Finally, as is shown in Fig 5, the S_{SF − ER} and the average of the S_{SF − SF} and S_{ER − ER} are closed. We hope these findings will be useful to construct robust functionally identical coupled networks.